EC2 SUPPLEMENT: PAPERBACK
EDITION OF 2001
Richard P. Stanley
version of 14 May 2016
Here I will maintain supplementary material for Enumerative Combinatorics, volume 2 (paperback edition of 2001). This will include errata,
updated references, and new material. I will be continually updating this
supplement.
Note. References to math.CO refer to the combinatorics section of the
xxx Mathematics Archive at xxx.lanl.gov/archive/math. A front end site for
math.CO is front.math.ucdavis.edu/math.CO.
• p. 6, line 10. Change situtations to situations.
• p. 11, line 3. Change Ec (n) to Ec (x).
• p. 18, line 3. Change (n)2 to n(n − 2).
• p. 20, line 9. Change Z(Sn ) to Z̃(Sn ).
• p. 24, line 4 (after figure). Change limn→∞ to limk→∞ .
• p. 25, line 5. Change ⊆ to ∈.
• p. 33, line 5–. Change ord(τk ) to ord(τj ).
• p. 34, Lemma 5.3.9. Delete the first sentence, viz., “Let w ∈ A∗ .”
• p. 35, line 10. Change w ∈ B ∗ to w ∈ Br∗ .
• p. 35, line 8–. Change A to A.
• p. 36, lines 15–16. Change “beginning with a 1” to “ending with a −1”.
• p. 36, line 1–. Insert + · · · before =. (The left-hand side is an infinite
sum.)
1
(2)
• p. 51, line 9–. Change Qi = Πi
ple 5.5.2(d) for r = 2”.
to “when Qi is given by Exam-
• p. 59, line 9. Change “Since the rows” to “Since the columns”.
• p. 59, line 13. Change “Because the columns” to “Because the rows”.
• p. 62. Example 5.6.12, line 5. Change “modulo n” to “modulo 2n ”.
• p. 63, line 12. Change “sequence” to “sequences”.
• p. 65, line 8. Change “Theorem” to “Lemma”.
• p. 87, equation (5.111). We need to add the further condition that
pn (0) = δ0n . Otherwise, for instance, the polynomials pn (x) = (1 + x)n
d
but fail to satisfy (i)–(iii).
satisfy (iv) with Q = dx
• p. 97, Exercise 5.51. It is not true that (ii) implies (i), e.g., when
C(x) = c. One needs to add the hypothesis that [x]C(x) 6= 0, so that
(C(x) − c)h−1i exists. Substituting xC(B(x)) for x in (ii) yields
xC(B(x))/C(A(xC(B(x)))) = x,
so C(B(x)) = C(A(xC(B(x)))). Substituting B(x)h−1i for x yields
C(x) = C(A(B(x)h−1i C(x))). Subtract c from both sides and apply
(C − c)h−1i to get x = A(B(x)h−1i C(x)). Applying Ah−1i to both sides
gives (i). This argument is due to Daniel Giaimo and Amit Khetan
and (independently) to Yumi Odama.
√
√
• p. 101, line 3. Change J0 [(2 − t)/ t − 1] to J0 −t (2 − t)/(1 − t) .
• p. 102, Exercise 5.71. It would be better not to specify the degree d of
G, since (as stated in the solution) d = λ1 .
• p. 103, Exercise 5.74(d). Replace the first two sentences with: Show
that all vertices have the same outdegree d. (By (c), all vertices then
also have indegree d.)
• p. 103, Exercise 5.74(f). For further information, see F. Curtis, J. Drew,
C.-K. Li and D. Pragel, J. Combinatorial Theory (A) 105 (2004), 35–
50, and the references given there.
2
• p. 108, Exercise 5.7(a), line 7. Change b2n to b2n−k .
• p. 137, Exercise 5.45, line 1. Change kxy k to (k + 1)xy k .
• p. 137, Exercise 5.45, line 4. Change this equation to
y = x + 2xy + 3xy 2 + · · · =
x
.
(1 − y)2
• p. 139, Exercise 5.47(c), line 7. A direct combinatorial proof was given
by M. Bousquet-Mélou and G. Schaeffer, Advances in Applied Math.
24 (2000), 337–368.
• p. 147, Third Solution. The first two lines should be: Equation (5.530)
can be rewritten (after substituting n + k for n)
1
(n + k)[x ]
k
n
F h−1i (x)
x
k
n
= [x ]
x
F (x)
n+k
.
(5.140)
• p. 162, lines 13– to 12–. Change “Thus any algebraic power series, as
defined in Definition 6.1.1” to “Thus any algebraic function, i.e, any
solution η to (6.2)”.
• p. 175, line 1. Change {9, 11} to {9, 14}.
• p. 175, line 2. Change x11 to x14 .
• p. 175, line 4. Change v 11 to v 14 .
• p. 175, line 11. Change Theorem 5.4.1 to Theorem 5.4.2.
• p. 175, line 2–. Change k ∈ K to k ∈ Z.
• p. 176, line 16. Change intesect to intersect.
• p. 176, line 4–. Change (n + 2)-gon to (n + 1)-gon.
• p. 192, line 9–. Change u(0) = 0 to v(0) = 0.
3
• p. 192, lines 8– to 7–. The example v = log(1 +p
x2 ) − 1 is confusing
since v(0) 6= 0. Nevertheless the series u(v(x)) = log(1 + x2 ) is welldefined formally since we can write
r
p
log(1 + x2 )
log(1 + x2 ) = x
.
x2
It would have been more accurate to define
log(1 + x2 )
− 1.
x2
The same remarks apply to Exercise 6.59.
v(x) =
• p. 212, line 1. The statement that Catalan number enumeration originated with Segner and Euler in 1760 (or actually 1758/59 in the cited
references) is inaccurate. The enumeration of polygon dissections was
stated by Euler in a letter to Goldbach in 1751. This letter is printed in
P.-H. Fuss, Correspondance Mathématique et Physique, Tome. 1, Acad.
Sci. St. Petersburg, 1843; reprinted in The Sources of Science, No. 35,
Johnson Reprint Corporation, New York and London, 1968, pp. 549–
552.
• p. 212, lines 16–17. I have forgotten the source for the statement that
Netto was the first to use the term “Catalan number.” Can anyone
provide a reference?
• p. 219, Exercise 6.16. A combinatorial proof was first given by R.
Sulanke, Electronic J. Combinatorics 7(1), R40, 2000. A sharper result
was subsequently proved combinatorially by D. Callan, A uniformly
distributed parameter on a class of lattice paths, math.CO/0310461.
• p. 221, Exercise 6.19(j).This problem appeared as Problem A5 on the
2003 William Lowell Putnam Mathematical Competition. Many participants found the following bijection with 6.19(i) (Dyck paths from
(0, 0) to (2n, 0)): Let D be a Dyck path from (0, 0) to (2n, 0). If D
has no maximal sequence of (1, −1) steps of even length ending on the
x-axis, then just prepend the steps (1, 1) and (1, −1) to the beginning
of D. Otherwise let R be the rightmost maximal sequence of (1, −1)
steps of even length ending on the x-axis. Insert an extra (1, 1) step at
the beginning of D and a (1, −1) step after R. This gives the desired
bijection.
4
• p. 224, item ii, line 5. Change S(w) = w to S(w) = 12 · · · n.
• p. 228, item iii, line 3. To be precise, the displayed sequences should
have the intial and final 1’s deleted.
• p. 230, Exercise 6.21(b), line 3. Change 5.3.11 to 5.3.12
• p. 230, Exercise 6.23. Brian Rothbach has pointed out that this problem can be given another stipulation: serieshelpmate in 20. By parity considerations (Black cannot lose an odd number of moves with a
knight) the solution is the same as before except Black plays Pa7-a6
instead of Pa7-a5. Thus the number of solutions is C10 = 16796.
• p. 233, Exercise 6.30, line 3. It would be less ambiguous to change “this
exercise” to “that exercise”.
• p. 235, Exercise 6.34, line 4. At the end of the line it should be mentioned that the polynomial g(Ln , q) of Exercise 3.71(f) is a further
q-analogue of Cn . An additional reference for this polynomial is R.
Stanley, J. Amer. Math. Soc. 5 (1992), 805–851 (Prop. 8.6).
• p. 236, Exercise 6.34(b,c). While (b) is correct as stated (in the paperback edition of 2001), it would be best to change q n cn (q) on line 6 to
cn (q) (as it was in the hardcover edition of 1999) and “nonnegative” on
line 8 to “nonpositive”. In this way part (c) remains valid. If part (b)
is kept as it is, then change cn (t; q) in line 4 of part (c) to q n cn (t; q).
• p. 238, Exercise 6.38(d), line 1. Change (n, n) to (n, 0).
• p. 239, Exercise 6.39(h). A period is missing at the end of the sentence.
• p. 241, Exercise 6.41, line 1. Change S 2 (w) = w to S 2 (w) = 12 · · · n.
• p. 247, Exercise 6.59. See the item above for p. 192, lines 8– to 7–.
• p. 250, Exercise 6.3, line 3. Replace “r = s +
“r cannot be a negative integer”.
1
2
for some s ∈ Z” with
t n
P
• p. 250, Exercise 6.3, paragraph 3. The earliest proof that n≥0 2n
x
n
isn’t algebraic for any t ∈ N, t > 1, appears in the paper P. Flajolet,
Theoretical Computer Science 49 (1987), 283–309 (page 294). Flajolet
5
P
shows that if
an xn is algebraic and each an ∈ Q, then an satisfies an
asymptotic formula
m
β n ns X
Ci ωin + O(β n nt ),
an =
Γ(s + 1) i=0
where s ∈ Q−{−1, −2, −3, . . . }, t < s, β is a positive algebraic number,
and the Ci and ωi are algebraic with |ωi | = 1. A simple application of
t
Stirling’s formula shows that if an = 2n
, then an does not have this
n
asymptotic form when t ∈ N, t > 1.
• p. 257, Exercise 6.19(k). Update the reference to J. Integer Seq. 4
(2001), Article 01.1.3; available electronically at
http://www.research.att.com/∼njas/sequences/JIS.
• p. 258, Exercise 6.19(s), line 1. Change ai to ai − 1.
• p. 260, Exercise 6.19(ee), lines 10–12. The statement that the first
published proof of the enumeration of 321-avoiding permutations is due
to D. G. Rogers is inaccurate. Knuth provided such a proof in 1973
in the reference given in the first paragraph of the solution. Moreover,
a bijective proof was found by D. Rotem, Inf. Proc. Letters 4 (1975),
58–61.
• p. 269, Exercise 4.23. Change the rating to [5].
• p. 274, line 2. Change “D. Vanquelin” to “B. Vauquelin”.
• p. 278, Exercise 6.53, line 3. Change Q(x) = x − 2 to Q(x) = −x − 2.
• p. 281, Exercise 6.60. An elegant proof based on Gröbner bases was
given by Chris Hillar, Proc. Amer. Math. Soc. 132 (2004), 2693–2701.
• p. 282, Exercise 6.63(b), line 2. Change 1847 to 1848.
• p. 293, lines 11-13. Replace “, and such that the . . . exist.)” with a
period. (The deleted condition automatically holds.)
• p. 295, Figure 7-3. In the expansion of h41 , the coefficient of m41 should
be 2.
6
• p. 298, line 10–. Change “if follows” to “it follows”.
• p. 301, line 7. Change 1.1.9(b) to 1.9(b).
• pp. 314–315, proof of Proposition 7.10.4. Change λ to λ/µ throughout
proof.
• p. 317, line 12–. Change “clearly impossible” to “clear”.
• p. 329, line 15–. Change x’s to X’s.
• p. 336, line 7 (counting the displayed tableau as a single line). Change
7.8.2(b) to 7.8.2(a).
• p. 346, line 3–. Change “forms a border strip” to “forms a border strip
or is empty”.
• p. 346, line 1–. Change λi /λi+1 to λi+1 /λi .
• p. 348, line 9. Change χλ (µ) to χλ (µ).
• p. 352, line 2 of proof of Proposition 7.18.1. Change
X
zλ−1 f (λ)pµ to
µ
X
zλ−1 f (λ)pλ .
λ
• p. 354, line 5. Change “a integral” to “an integral”.
• p. 355, line 4. Add a period after “nonnegative”.
• p. 359, line 6. Change the subscript αS to co(S).
• p. 364, line 1. Change e(D(T )) to e(co(D(T ))).
• p. 370, line 3 of second proof. Change 1.22(d) to 1.23(d).
• p. 377, line 7; p. 378, line 8; page 378, line 10–. Change π ∈ B(r, c, t)
to π ⊆ B(r, c, t).
• p. 379, line 5–. Insert π after the first “partition”, and change λ∗ to
π∗.
• p. 379, line 4–. Change “similary” to “similarly”.
7
• p. 383, line 9. Change “D(w) = T 0 and D(w−1 ) = T ” to “D(w) =
D(T 0 ) and D(w−1 ) = D(T )”.
• p. 395, line 10–. Change “Burnside’s theorem” to “Burnside’s lemma”.
• p. 416, line 7–. Change uit+2 to ujt+2 .
• p. 418, line 7. Change “subsequences” to “subsequence”.
• p. 419, line 16. Change “was” to “is”.
• p. 421, line 9–. Insert “a” after “such”.
• p. 421, lines 8– to 7–. Change “second statement of Theorem A1.1.4”
to “first assertion of Theorem A1.1.6”.
• p. 422, line 3. Change A1.1.4 to A1.1.6.
• p. 424, line 11. Delete “by”.
• p. 426, line “tableaux in (A1.137)” to “tableau defined by (A1.137)”.
• p. 439, line 7. Delete comma after 156.
• p. 442, Theorem A2.4, line 9. Change “Hence” to “Moreover,”.
• p. 443, line 11. Change
char ϕ = (x1 · · · xn )−1 = (x1 · · · xn )−1 s∅
to
n−1
−1
−1 1
char ϕ = x−1
1 + · · · + xn = (x1 · · · xn ) s
• p. 444, line 12. Delete “char”.
• p. 444, line 11–. Change “given by (A2.156)” to “generated (as a Calgebra) by (A2.156)”.
• p. 447, line 3–. Change s1 (xλ1 i ) to s1 (xλ1 i , xλ2 i , . . . ).
8
• p. 451, Exercise 7.13(a). This exercise is stated incorrectly. For instance, K777,6654 = 1, contrary to the statement of the exercise. One
way to state the correct result is as follows. Let the parts of λ0 be given
by as follows. Let the parts of λ0 be given by
λ01 = · · · = λ0n1 > λ0n1 +1 = · · · = λ0n2 > λ0n2 +1 = · · ·
> λ0nk−1 +1 = · · · = λ0nk > 0.
Define λ(j) = (λ0nj−1 +1 , . . . , λ0nj ) (with n0 = 0), so λ(j) is a partition of
rectangular shape. Let µ be a partition with |µ| = |λ|, and let
µ(j) = (µnj−1 +1 , . . . , µnj ).
Then Kλµ = 1 if and only if λ ≥ µ (dominance order) and
(i) |λ(j) | = |µ(j) | and λ(j) ≥ µ(j) for all j.
(ii) For all 1 ≤ j ≤ k either 0 ≤ µ0nj−1 +1 − λ0nj−1 +1 ≤ 1 or 0 ≤
λ0nj − µ0nj ≤ 1.
• p. 452, line 6. Change “k times” to “n times”.
• p. 452, Exercise 7/16(a), line 5. Change ci−j + ci+j to Change ci−j ici+j .
• pp. 452–453, Exercise 7.16(b,e). The formulas for yi (n) and ui (n) have
been extended to i ≤ 6 by F. Gascon, Fonctions de Bessel et combinatoire, Publ. LACIM 28, Univ. du Québec à Montréal, 2002 (page 75).
In particular,
y6 (2n) = 6(2n)!
n
X
k=0
(10n − 13k + 8)Ck+1
,
(n − k + 2)! (n − k)! (k + 4)! k!
where Ck+1 denotes a Catalan number.
• p. 459, Exercise 7.30(c), line 4. Change d − 1 to d.
• p. 460, Exercise 7.37. For further information on expanding a2δ in terms
of Schur functions, see
http://www.phys.uni.torun.pl/∼bgw/vanex.html.
9
• p. 461, Exercise 7.42, line 2. Change sλ̃ (y) to sλe0 (y).
• p. 466, line 3–. Change (λi − 1)!(λ0i − 1)! to (λi − i)!(λ0i − i)!.
• p. 467, Exercise 7.55(b). Let f (n) be the number of λ ` n satisfying
(7.177). Then
(f (1), f (2), . . . , f (30)) = (1, 1, 1, 2, 2, 7, 7, 10, 10, 34, 40, 53, 61,
103, 112, 143, 145, 369, 458, 579, 712, 938, 1127,
1383, 1638, 2308, 2754, 3334, 3925, 5092).
The problem of finding a formula for f (n) was solved by Arvind Ayyer,
Amritanshu Prasad, and Steven Spallone, arXiv:1604.08837.
• p. 467, Exercise 7.59. In order for the bijection λ 7→ (λ0 , λ1 , . . . , λp−1 )
given in the solution to part (e) (page 517) to be correct, it is necessary
to define a specific indexing of the terms of Cλ . Namely, index a term
a by ci if i = i1 − i0 , where i1 is the number of 1’s weakly to the left
of a, and i0 is the number of 0’s strictly to the right of a (so if a = 1,
then this contributes to i1 ). The sequence becomes · · · c−2 c−1 c0 c1 c2 · · ·
as before, so it suffices to define the indexing by letting the first 1 be
ci0 −1 , where i0 is the number of 0’s following this 1.
Example. If λ = (4, 3, 3, 3, 1), then Cλ = · · · 0010110001011 · · · . The
first 1 in this sequence is c1−4 = c−3 . On the other hand, if λ =
(3, 3, 3, 2, 2, 1), then Cλ = · · · 0010100100011 · · · . Now the first 1 is
c1−6 = c−5 .
• p. 468, Exercise 7.59(e), line 3. Change Y k to Y p .
• p. 469, Exercise 7.61, line 2. Change “0 or 1” to “0 or ±1”.
• p. 474, Exercise 7.70, line 3. Under the second summation sign insert
a space between “in” and Sn .
• p. 485, line 3–. The
√ asymptotic formula
√ for a(n) should be multiplied
by a factor of 1/ 3π. The factor 1/ π was included
√ by Wright and
omitted here by mistake. The additional factor 1/ 3 was omitted by
Wright, though his proof makes it clear that it should appear. See L.
Mutafchiev and E. Kamenov, math.CO/0601253.
• p. 491, Exercise 7.9, line 1. Insert ελ before aλµ eλ .
10
• p. 492, Exercise 7.11. Change
j
`(µ)−1
to
`(µ)−1
j
(three times).
• p. 493, Exercise 7.13(a). For another proof, see A. N. Kirillov, Europ.
J. Combinatorics 21 (2000), 1047–1055, arXiv:hep-th/9304099 (Prop.
2.2).
• p. 494, line 4. Change 169–172 to 175–177.
• p. 494, Figure 7-20. Change the labels R1 h6, R1 h5, and R2 h6 to R1 a6,
R1 a5, and R2 a5, respectively.
Q
• p. 496, equation (7.199). Change (mi (λ)!)−1 to [ i (mi (λ)!)−1 ].
• p. 498, Exercise 7.22(h), line 7. Update the Fomin and Greene reference
to Discrete Math. 193 (1998), 179–200.
• p. 500, displayed tableaux near end of Exercise 7.24. The tableaux T8
and T9 are missing the element 8 to the right of 3. Also, the {3, 10}
under T9 should be under T10 .
• p. 502, Exercise 7.27, first displayed equation. Change (n)m to (n)n−m .
• p. 505, Exercise 7.32(a). Stembridge’s more general result appears in
“Computational aspects of root systems, Coxeter groups, and Weyl
characters,” in Interactions of Combinatorics and Representation Theory, MSJ Memoirs 11, Math. Soc. Japan, Tokyo, 2001, pp. 1–38 (Theorem 7.4).
• p. 514, Exercise 7.47(m), lines 1–3. Update the reference to R. Stanley,
Discrete Math. 193 (1998), 267–286.
• p. 514, Exercise 7.48(b), lines 2–4. Update the reference to R. Simion
and R. Stanley, Discrete Math. 204 (1999), 369–396.
• p. 514, line 1–, and p. 515, line 1. “Ibid.” refers to the reference in the
item above, not to the previous reference in the book.
• p. 515, line 3. “the reference” refers to the reference two items above,
viz., R. Stanley, Electron. J. Combinatorics 3, R6 (1996), 22 pp.
• p. 515, Exercise 7.49. Update this reference to C. Lenart, J. Algebraic
Combin. 11 (2000), 69–78.
11
• p. 516, line 8. Change (λi − 1)!(λ0i − 1)! to (λi − i)!(λ0i − i)!.
• p. 517, Exercise 7.59(e), line 3. Change Y k to Y p .
• p. 517, Exercise 7.59(e), line 9. Change Y∅ to Yp,∅ , and change Y k to
Y p.
• p. 518, Exercise 7.59(h), line 1. Change Y∅ to Yp,∅ , and change Y k to
Y p.
• p. 518, Exercise 7.59(h), line 2. Change Y k to Y p (three times).
• p. 518, Exercise 7.59(h), line 3. Change Y k to Y p .
P
P
• p. 520, line 3–. Change n≥0 h2n+1 t2n+1 to n≥0 (−1)n h2n+1 t2n+1
• p. 535, lines 7–10. Replace the sentence “No proof . . . are known.”
with “A bijective proof of the unimodality of sλ (1, q, . . . , q n ) was given
by A. N. Kirillov, C. R. Acad. Sci. Paris, Sér. I 315 (1992), 497–501.”
• p. 537, Exercise 7.78(f), line 6. Change sµ (x) to sµ (y) and sν (x) to
sν (z).
• p. 576, line 7. Change work to word.
• p. 584, fifth item. Update the reference to K. S. Kedlaya, Proc. Amer.
Math. Soc. 129 (2001), 3461–3470.
• p. 584, eleventh item. Update this reference to J. H. Przytycki and A. S.
Sikora, J. Combinatorial Theory(A) 92 (2000), 68–76, math.CO/9811086.
• p. 584, first item. Add the additional reference P. J. Larcombe and P.
D. C. Wilson, Congr. Numerantium 149 (2001), 97–108.
• p. 584, third item. Update the reference to M. Haiman, J. Amer. Math.
Soc. 14 (2001), 941–1006; math.AG/0010246.
• p. 585, ninth item. Update the reference to P. Hersh, J. Combinatorial
Theory (A) 97 (2002), 1–26.
• p. 580. Change “Valquelin, D.” to “Vauquelin, B.”.
12